October 2, 2018 EFI-18-14

(2,2) Geometry from Gauge Theory

Joao Caldeiraa, Travis Maxfieldb, and Savdeep Sethia

a Enrico Fermi Institute & Kadanoff Center for Theoretical Physics

University of Chicago, Chicago, IL 60637, USA

bCenter for Geometry and Theoretical Physics, Box 90318

Duke University, Durham, NC 27708, USA

Email: jcaldeira@uchicago.edu, travis.maxfield@duke.edu, sethi@uchicago.edu

Abstract

Using gauge theory, we describe how to construct generalized Kahler geometries

with (2, 2) two-dimensional supersymmetry, which are analogues of familiar examples

like projective spaces and Calabi-Yau manifolds. For special cases, T-dual descriptions

can be found which are squashed Kahler spaces. We explore the vacuum structure

of these gauge theories by studying the Coulomb branch, which usually encodes the

quantum cohomology ring. Some models without Kahler dual descriptions possess

unusual Coulomb branches. Specifically, there appear to be an infinite number of

supersymmetric vacua.

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Contents

1 Introduction 2

2 Dual Descriptions 7

2.1 Holomorphic data and kinetic terms . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Worldsheet duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Anomaly and conditions for conformality . . . . . . . . . . . . . . . . . . . . 11

3 A Collection of Models 14

3.1 Bounding D-terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 One U(1) action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Two U(1) actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.3 Compact and conformal models? . . . . . . . . . . . . . . . . . . . . 18

3.2 The Kahler picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 One U(1) action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 Two U(1) actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 More general fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 A comment on squashed Calabi-Yau . . . . . . . . . . . . . . . . . . 23

3.4 Exponential couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.1 One U(1) action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.2 Several U(1) actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5 Unifying constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 The Quantum Cohomology Ring 30

4.1 Coulomb branch vacua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Double trumpet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Exponential models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4 An explicit example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.5 Unified structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A (2, 2) Superspace Conventions 39

A.1 Basic conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

B Dualization 41

B.1 Component picture and periodicity . . . . . . . . . . . . . . . . . . . . . . . 41

1

B.2 (2, 2) duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

C Central Charge and Anomaly Details 46

1 Introduction

This work concerns sigma models in two dimensions with target space M and local coor-

dinates φ. Ignoring fermions, the bosonic sigma model action in a target space patch takes

the form

S =1

4πα′

∫d2x√hhαβGij∂αφ

i∂βφj + i

∫φ∗(B), (1.1)

where h is the two-dimensional worldsheet metric, while G and B denote the target space

metric and 2-form B-field. Requiring extended worldsheet supersymmetry means introduc-

ing fermions and also restricting the target spaceM. Of particular interest are models with

chiral (0, 2) worldsheet supersymmetry, suitable for the heterotic string, and models with

non-chiral (2, 2) supersymmetry suitable for both the heterotic and type II strings.

Target spaces which are compatible with (2, 2) supersymmetry are called generalized

Kahler spaces [1–3]. There are two basic issues one might try to address. The first is

classifying the geometric structures required forM to admit (2, 2) supersymmetry, and the

corresponding implications for superspace constructions. There has been a great deal of

progress along these lines starting with [1]. For general (2, 2) non-linear sigma models, the

basic needed superspace ingredients are chiral, twisted chiral and semi-chiral superfields [4].

See [5] for a recent discussion of the defining data for classes of (2, 2) models.

The second issue is the question of constructing classes of (2, 2) target spaces. This ques-

tion has a somewhat different flavor because acceptable target spaces can include ingredients

that require a physical explanation; for example, spaces with orbifold singularities, particu-

larly those with discrete torsion, brane sources, or the use of stringy worldsheet symmetries

like T-duality in patching conditions.

The simplest examples of (2, 2) sigma models have Kahler target spaces M. Imposing

conformal invariance further restricts M to a Calabi-Yau space. Once one reaches Calabi-

Yau 4-folds, there are believed to be an enormous number of such spaces with lower bound

estimates of O(10755) [6], and a recent Monte-Carlo based estimate of O(103000) [7]! On top

of this geometric degeneracy is the usual enormous number of choices of flux, estimated in

one case to be O(10272,000) [8]. Somewhat surprising is the realization that a large fraction

of these Calabi-Yau spaces admit elliptic fibrations and even K3-fibrations [9].

2

Duality between the heterotic string and K3-fibered F-theory flux vacua, built from

Calabi-Yau 4-folds, suggests that there should exist an enormous number of worldsheet

string geometries with non-vanishing H = dB [10]. These are not Calabi-Yau manifolds

but rather a kind of torsional background compatible with (0, 2) worldsheet supersymmetry.

The expected number of such geometries should dwarf the number of currently known

Calabi-Yau 3-folds. Yet very few compact examples are known. Unlike the case of Kahler

target spaces, there are few if any systematic constructions of flux geometries with H 6= 0.

We are missing tools like algebraic geometry which might provide us with large classes of

such spaces.

This picture motivates us to move away from the familiar Kahler geometries visible

under the lamp post, and search for the new ingredients and structures needed to describe

more generic string geometries with non-vanishing H. Along the way, we will learn more

about the physics of NS-branes and anti-branes. By an NS-brane we mean a localized

magnetic source for B such that the charge is non-vanishing,∫C3

H 6= 0, (1.2)

where C3 encloses the brane. The sign of the charge distinguishes a brane from an anti-

brane. When the sigma model (1.1) is conformal and can serve as a classical string back-

ground, these NS-branes are the familiar NS5-branes. However, the definition (1.2) applies

to both gapped and conformal sigma models.

The approach we will take is to generalize the gauged linear sigma model (GLSM) con-

struction described by Witten [11]. Our generalization is motivated by the (0, 2) construc-

tions described in [12–16], and specifically [17]. We will provide analogous constructions for

models with (2, 2) supersymmetry. The enhanced (2, 2) supersymmetry makes a far larger

set of tools available for analysis. While the most general model with (2, 2) supersymme-

try involves semi-chiral superfields, in this work we will restrict our discussion to models

constructed from chiral superfields Φ satisfying

D+Φ = D−Φ = 0, (1.3)

and twisted chiral superfields Y satisfying

D+Y = D−Y = 0. (1.4)

Our conventions are described in Appendix A. We will also only consider abelian gauge

theories. In the usual Kahler setting, this corresponds to considering toric spacesM. Gen-

3

eralizing these constructions by considering non-abelian gauge theories, and by including

semi-chiral representations is likely to be interesting.

The main new ingredient over the original work of [11] is the inclusion of periodic

superfields,

Y ∼ Y + 2πi. (1.5)

Such periodic fields appear in mirror descriptions of (2, 2) and (0, 2) GLSM theories [18–20],

and in earlier GLSM constructions for torsional target spaces [21,22,12]. The superfield Y

can be used to build field-dependent Fayet-Iliopoulos (FI) couplings,∫d2xdθ+dθ−Y Σ, (1.6)

where Σ is the field strength for a vector superfield. This coupling leads to torsion in the

target space. As we will see later, including more couplings respecting the periodicity (1.5),

like twisted superpotentials involving eY , gives interesting physical models.

T-dual

Mirror

Cigar

Liouville

Re(y)

Im(y)

Figure 1: The basic trumpet geometry for an NS-brane and its dual realizations.

We can sketch the basic building blocks for our constructions. With one U(1) gauge

field and a field-dependent FI coupling (1.6), the target space M is non-compact. This

ingredient is depicted in Figure 1. The picture in Y variables is the ‘trumpet’ geometry,

4

while a T-dual description gives the cigar geometry. Finally a mirror description gives the

Liouville theory, which involves a potential energy coupling rather than pure geometry. The

blowing up of the Im(Y ) circle is the hallmark of the brane in Y variables. This target

space is a conformal field theory if one includes an appropriate varying dilaton field. The

equivalence between the (2, 2) cigar and Liouville descriptions was argued by Giveon and

Kutasov [23, 24], building on earlier work and conjectures [25, 26]. A GLSM derivation of

the equivalence was provided by Hori and Kapustin [27]. The three pictures of the same

physical system make clear the need to include T-duality – the equivalence between large

and small circles in string theory – in the patching conditions for the target geometry

M. The trumpet is better described in terms of the cigar geometry near the locus where

the circle blows up, while the cigar is better described by the Liouville theory when the

asymptotic circle becomes small. In addition, the need for the Liouville description makes

clear that we must allow potentials as well as metrics and B-fields when discussing more

general notions of string geometry. This is quite reminiscent of the structure seen in hybrid

Landau-Ginzburg phases; see, for example [28,29].

With multiple abelian gauge fields, the picture becomes richer. Instead of a semi-

infinite trumpet, we can build finite-sized cylindrical fixtures. A product of toric varieties

is typically fibered over each fixture with varying Kahler parameters. This is schematically

depicted in Figure 2. The interpretation of this geometry is that one end of the fixture

supports a wrapped brane while the other end supports an anti-brane. In a precise sense,

these geometries are torsional dual descriptions of compact squashed toric varieties like

projective spaces, introduced by Hori and Kapustin [27]. This duality, which generalizes

the standard relation between NS5-branes and ALF spaces, is described in section 2. As we

will see below, it is useful to have both descriptions of the same physical system in order

to explore generalizations.

Generic models involving this collection of ingredients cannot, however, be dualized to

purely Kahler spaces; they are intrinsically torsional. We discuss two flavors of such models.

The first flavor involves allowing the complex structure parameters of a fibered space to

vary with Y . In a sense, this is the mirror version of the Kahler fibration of Figure 2. This

leads to generalized Kahler spaces of schematic form depicted in Figure 3. We describe

examples of such models in section 3, and discuss the appearance of flat directions when

one attempts to construct conformal models – the analogues of Calabi-Yau spaces – as

complete intersections in the torsional analogues of toric varieties. We find it plausible that

the flat directions we persistently find when a fibered toric space shrinks to zero size reflect

5

Re(y)

Im(y)

Figure 2: A product of toric spaces fibered over the cylindrical fixture.

the non-perturbative space-time physics supported on the NS-branes. This would match

other examples where space-time non-perturbative physics, like the appearance of enhanced

gauge symmetry at an ALE singularity, is reflected by a new branch in the associated gauged

linear sigma model.

The other generalization leads to wilder structures. Instead of simply allowing couplings

like (1.6) which preserve the U(1) isometry that shifts the imaginary part of Y , we consider

more general field-dependent FI couplings,∫d2xdθ+dθ−f(Y, eY )Σ. (1.7)

The inclusion of interactions like this in otherwise topological interactions is familiar from

N = 1 and N = 2 D = 4 gauge theory, where superpotential or prepotential interactions

can be generated by instantons or strong coupling effects. In D = 2, we can include such

couplings in the ultraviolet model and such models are described in section 3.4. The classical

vacuum equations have interesting new properties. We comment on some puzzling but

interesting aspects of the quantum vacuum structure on the Coulomb branch of these models

in section 4. These Coulomb branch vacua are usually related to quantum cohomology rings

of the target space M. We find that the inclusion of these more general U(1) breaking

6

couplings leads to an infinite number of discrete Coulomb branch vacua, which is in sharp

contrast to the finite number of vacua found in GLSMs describing Kahler spaces. We also

find an analogue of the quantum cohomology ring for a class of generalized Kahler examples.

Finally, it is worth noting that the way in which the spaces are constructed has a

flavor similar to a recent construction of G2 spaces at the level of geometry and conformal

field theory [30–32]. It would be very interesting if this gauged linear construction can be

generalized to produce target geometries with G2 holonomy.

Re(y)

Im(y)

Figure 3: The more general case with both complex and Kahler parameters fibered over the

fixture.

2 Dual Descriptions

We will consider models built from chiral superfields Φ with charge Q under an abelian

gauge group. All basic conventions are found in Appendix A.1. These superfields contain

one complex scalar φ. Under a gauge transformation with chiral superfield parameter Λ,

the vector superfield V for the gauge symmetry and Φ transform as follows:

V → V +i

2(Λ− Λ), Φ→ eiQΛΦ. (2.1)

The field strength superfield defined in (A.8) is denoted Σ. It contains one complex scalar

σ. Following the notation of [27], we also consider chiral superfields P which are shift

7

charged under the gauge group:

P → P + i`Λ, P ∼ P + 2πi. (2.2)

with ` ∈ Z. The chiral superfield P also contains one complex scalar p. We will write the

kinetic terms of P as

SPkin =b

32π

∫d2xd4θ

(P + P + 2`V

)2, (2.3)

allowing an overall factor b. We could allow the periodicity to vary as well, introducing

an additional real parameter δ and defining P such that P ∼ P + 2πiδ. However, we can

always scale P to have δ = 1, absorbing the change into b and `. This is what we will do in

this work. The last ingredient is twisted chiral neutral superfields Y with periodicity given

in (1.5) containing a complex scalar y.

Now this is a highly asymmetric treatment of chiral versus twisted chiral superfields.

Mirror symmetry exchanges the two kinds of constrained superfield, and it was recognized in

very early attempts to prove mirror symmetry that a more symmetric treatment might prove

helpful. Following the terminology of Morrison and Plesser [33], we introduce superfields

for a twisted GLSM. These fields are denoted by a ‘hat’ so Φ is a twisted chiral superfield

charged under a vector superfield V with chiral field strength Σ. The chiral superfields Y

are neutral and periodic, just like their twisted chiral cousins in (1.5),

Y ∼ Y + 2πi. (2.4)

These are similar to the fields P in (2.2), except we will always take fields Y to not transform

under the gauge group while P is shift-charged. As we will see in section 3.3, all these field

types are very natural for describing general models.

2.1 Holomorphic data and kinetic terms

We can group the collection of fields into uncharged periodic fields (Y, Y ), charged fields

(Φ, Φ, P, P ) and gauge fields (V, V ) with field strengths (Σ, Σ). The data over which we

have the most control under RG flow are holomorphic couplings. These include the super-

potential,

SW =1

8π

∫d2xdθ+dθ−

{−itΣ− kΣY +W (Φ, P, eY , Σ) + c.c.

}, (2.5)

where t determines the FI parameter via (A.16), k is an integer in order to be compatible

with the Y periodicity, and W is gauge-invariant and single-valued. There is an analogous

8

twisted chiral superpotential with form,

SW =1

8π

∫d2xdθ+dθ−

{−itΣ− kΣY + W (Φ, P , eY ,Σ) + c.c.

}, (2.6)

where again k ∈ Z and W is single-valued.1 It is clear from the dependence of the superpo-

tential on t and Y that we could choose to absorb the FI parameters t into a constant shift

of the fields Y , without affecting the Y kinetic terms. However, the FI parameters t will

typically be additively renormalized. While we could choose to absorb this renormalization

into a shift of Y , for clarity we choose to always keep t explicit in this work. The function

W may include terms such as the typical polynomials in the charged fields Φ, polynomials

including Φ as well as eP and eY , and other terms including only Y and Σ like eY Σ. We

will briefly explore these possibilities in what follows.

These are the obvious holomorphic couplings, but there is actually more potential holo-

morphic data hidden in the kinetic terms because of the periodic fields (Y, Y ). Let us focus

on the Y fields. A general kinetic term for Y takes the form,

SYkin = − 1

16π

∫d2xd4θ

(Y f1 + f2 + c.c.

), (2.7)

where the fi are gauge-invariant functions of the superfields, and f2 is single-valued. To

be sensible under the periodic identification (1.5), f1 must be annihilated by∫d4θ. This

constrains f1 to be a holomorphic function of the twisted chiral superfields; f1 can also

depend either holomorphically or anti-holomorphically on the chiral superfields, so that

D+f1 = 0 or D−f1 = 0. This holomorphic data is likely to prove interesting for non-linear

sigma models.

In a linear theory, we are typically interested in kinetic terms that are quadratic in the

fields. At the quadratic level, there are no direct couplings of chiral and twisted chiral fields

so the Y and Y fields do not kinetically mix. If there are n Y -fields then the choice of Y

kinetic term corresponds to a choice of metric and B-field for T n encoded in kµν

SYkin = − 1

16π

∫d2xd4θ kµνYµYν . (2.8)

The classical moduli space is the familiar n2-dimensional Narain moduli space,

O(n, n,R)

O(n)×O(n). (2.9)

1Actually the superpotential and twisted superpotential need not be classically gauge-invariant if one

includes∫d4θV V couplings, which appear in [33].

9

We will not worry about discrete identifications on the moduli space since those identifi-

cations are not generally preserved by interactions. A similar discussion applies to the Y

fields.

The charged and uncharged fields also do not mix kinetically at the level of quadratic

interactions. For a Φ field with charge Q, we simply assume canonical kinetic terms,

SΦkin =

1

16π

∫d2xd4θ Φe2QV Φ, (2.10)

whose component form appears in (A.9), and similarly for a charged Φ field.

2.2 Worldsheet duality

Here we present the duality dictionary we will subsequently use. More details are presented

in Appendix B. A (2, 2) chiral superfield P with periodicity given in (2.2) can be axially

charged, making its imaginary part a two-dimensional Stueckelberg field. The action is

simply,

S =b

32π

∫d2xd4θ

(P + P + 2V

)2, (2.11)

where b is a constant that will later be interpreted as a squashing parameter for the models

of section 3.2. This particular theory has a dual description in terms of a (2, 2) twisted

chiral superfield Y with the same periodicity, and with action:

Sd = − 1

16πb

∫d2xd4θ Y Y − 1

8π

∫d2xdθ+dθ− Y Σ + c.c.. (2.12)

This duality requires the coupling between Y and Σ found in (1.6), which will feature

heavily in this work.

Alternatively, a (2, 2) chiral Φ parametrizing C can also be dualized. The action for a

charged chiral takes the form,

S =1

16π

∫d2xd4θ |Φ|2e2V , (2.13)

and the dual Y is also a twisted chiral superfield with periodicity (1.5), and with action:

Sd =− 1

16π

∫d2xd4θ (Y + Y ) log(Y + Y )

−[

1

8π

∫d2xdθ+dθ−

(Y Σ + µe−Y

)+ c.c.

]. (2.14)

10

2.3 Anomaly and conditions for conformality

Given a (2, 2) gauged linear sigma model defined in the ultraviolet, it is usually a non-

trivial issue to decide whether or not it flows to a non-trivial conformal field theory. One

way to strengthen the case for a non-trivial infrared fixed point is to construct a candidate

N = 2 superconformal algebra in the ultraviolet theory. This boils down to finding a non-

anomalous right-moving R-symmetry current. In usual (2, 2) gauged linear sigma models

built using only chiral superfields, this is possible if the sum of the charges vanishes,∑i

Qia = 0, (2.15)

for all gauge symmetries. This means that the FI parameters of the theory will be invariant

under renormalization group flow, and that the curvature two-form for the non-linear model

found by symplectic quotient will be trivial in cohomology, so there is a Ricci-flat metric

in the same Kahler class. However, this condition is modified once we add P and Y fields.

In this section we will find the more general condition.

Let us start by defining our R-symmetries in the gauge theory. Under U(1)R, θ+ has

charge 1, while under U(1)L, θ− has charge 1. In a V-A basis, qθ+ = (1, 1), and qθ− = (1,−1).

Note dθ transforms as θ−1, so a chiral superpotential should have R transformation with

charges (2, 0) in that basis, while a twisted superpotential would have charges (0, 2). R-

invariance of the gauge field fixes qΣ = (0, 2); in turn, this implies that the FI twisted

superpotential tΣ has the correct R-charge. A chiral superfield with charge Q causes an

anomaly under U(1)A, since qψ+−qψ− = −2. This corresponds to a variation of the effective

action

δS =βQ(−2)

4π

∫d2xεµνFµν , (2.16)

where β is the transformation parameter. The same expression is valid for a twisted chiral

with charge Q under U(1)V . We can see from (2.14) and (A.14) that reproducing this

anomaly fixes the transformation under U(1)A for a field Y dual to a charged Φ to be a

shift charge of −2, so e−Y in (2.14) transforms like a twisted superpotential.

On the other hand, when Y is not dual to a charged Φ, its U(1)A transformation is not

fixed. Defining it generally to have a shift charge 2γ under U(1)A, or ±γ under U(1)R,L,

will cause the action (2.6) with W = 0 to vary as

δS =β(2kγ)

4π

∫d2xεµνFµν (2.17)

11

from the coupling (A.14). The updated condition for a non-anomalous U(1)R symmetry

with Y µ fields coupled to gauge fields Σa with coefficients kµa is then obtained by requiring

the sum of variations (2.16) and (2.17) to vanish,∑i

Qia −∑µ

γµkµa = 0 (2.18)

for some charges γµ.

What about our fields P with Stueckelberg couplings? At first sight, unlike Y they

do not have a coupling that would let us compensate an anomalous variation with a field

transformation. However, in many cases we can take a field Y and dualize it into P , so

the same condition must apply in both pictures. To understand how this works, note that

postulating a transformation y → y + iγβ means we are modifying the axial R-symmetry

current by

j+ = · · · − iγ

2∂+(y − y), j− = · · · − iγ

2∂−(y − y). (2.19)

If we dualize y into p, from (B.19), these current modifications become

j+ = · · · − iγ

2D+(p− p), j− = · · ·+ iγ

2D−(p− p). (2.20)

These are the current improvement terms constructed in [27]. These terms do not give p a

variation under U(1)A, but they modify the current conservation equation. In conclusion,

for each field pα with shift charges `αa under the gauge symmetries, we will also have a

parameter that modifies the condition for a non-anomalous R-symmetry into∑i

Qia −∑α

γα`αa −∑µ

γµkµa = 0. (2.21)

This is the condition we must work with in models with Y µΣa couplings. On the other

hand, note that a superpotential term of the form eYµΣ precludes any transformation of

that field Y µ under U(1)R, setting γµ = 0. This is because Σ alone must have the correct

transformation for a twisted superpotential as long as the FI coupling is non-trivial.

When we do expect a theory to flow to a non-trivial infrared fixed point, we can con-

struct the protected, right-moving chiral algebra in the ultraviolet whose central charge, in

particular, agrees with that of the infrared theory in the absence of accidents [34]. We will

construct the right-moving superconformal multiplet for a general class of models here.

12

A very general model constructed from the ingredients we have considered is the follow-

ing:

S =1

16π

∫d2xd4θ

(|Φi|2e2QiaVa +

bα2

(Pα + Pα + 2`αaVa

)2 − 1

bµ|Yµ|2 −

2π

e2a

|Σa|2)

− i

8π

∫d2xdθ+dθ− (ta − ikµaYµ) Σa + c.c. (2.22)

+1

8π

∫d2xdθ+dθ− W

(Φi, e

Pα)

+ c.c..

For the moment, let us ignore the superpotential term. This model has a protected, right-

moving superconformal multiplet of the form:

J 0−− = − 1

8πD−(e2Qi·V Φi

)e−2Qi·V D−

(e2Qi·V Φi

)− bα

8πD−(Pα + Pα + 2`α · V

)D−(Pα + Pα + 2`α · V

)(2.23)

− 1

8πbµD−YµD−Yµ −

1

4e2a

ΣaD−D−Σa.

This is the superconformal multiplet assigning R-charge 0 to the Φi fields. When we include

the superpotential, we will modify this multiplet by the addition of a flavor symmetry under

which the Φi rotate.

Classically, J 0−− satisfies

D+J 0−− = 0; (2.24)

however, there is a 1-loop anomaly that modifies this to

D+J 0−− =

γa4πD−Σa. (2.25)

Following [27], we determine the anomaly coefficient γa by point-splitting, the details of

which are found in Appendix C. The result is

γa =∑i

Qia. (2.26)

As in [27], a modified current can be defined that is 1-loop superconformal. Unlike in that

case, we will also take advantage of the freedom to include the Yµ fields in that modification.

In particular, we use

J−− = J 0−− +

γα8π

[D−, D−

] (Pα + Pα + 2`αaVa

)+

γµ

8π√bµ

[D−, D−

] (Yµ + Yµ

), (2.27)

13

where γα and γµ are chosen such that

γa = γα`αa + γµkµa. (2.28)

This is the same as (2.21).

Including the superpotential, J−− is no longer D+ closed. Instead,

D+J−− =1

4π((D−Φi) ∂i + (D−Pα + γαD−) ∂α)W

(Φi, e

Pα). (2.29)

We can partially remedy this by assuming the quasi-homogeneity of W and including in

J−− the flavor symmetry under which each chiral field Φi rotates with the corresponding

degree:

J−− → J−− + F−−, F−− =αi8πD−D−

(|Φi|2e2Qi·V

), (αiΦi∂i + βα∂α)W = W.

(2.30)

Then,

D+ (J−− + F−−) =1

4π(βα + γα)D−∂αW. (2.31)

Assuming the above is zero, i.e. βα + γα = 0 and we have a non-trivial infrared CFT,

we can find its central charge from the supercurrent we have just written. Details of this

calculation are found in Appendix C. The result is

c = 3

(∑i

(1− 2αi)−NU(1) +NP +NY + 2∑α

γ2α

bα+ 2

∑µ

γ2µ

bµ

). (2.32)

The first two terms are familiar from standard GLSM model building. The next two terms,

NP and NY are the number of P and Y fields, respectively, while the last two terms are

modifications due to the non-standard shift transformations required of the P and Y fields

in order to cancel all anomalies. In the next section, we will comment on specific examples

of models and their supercurrents and central charges, when possible.

3 A Collection of Models

We will now describe a series of examples of varying complexity that serve to illustrate

some of the possible target geometries described by this construction. Here we are only

concerned with the classical geometry that emerges from minimizing the potential energy

for a given GLSM. This corresponds to solving the D-term and F -term conditions, and

quotienting by the gauge group action. For the usual Kahler setting, solving the D-term

conditions and quotienting by the gauge group action defines a toric variety via symplectic

quotient. Further imposing F -term conditions gives an algebraic variety.

14

3.1 Bounding D-terms

3.1.1 One U(1) action

Turning on a single field-dependent FI-term, we work with the action

S =1

16π

∫d2xd4θ

(∑i

|Φi|2e2QiV − |Y |2 − 2π

e2|Σ|2

)− i

8π

∫d2xdθ+dθ− (t− ikY ) Σ + c.c. (3.1)

leading to a D-term potential imposing∑i

Qi|φi|2 = r − 2kRe(y). (3.2)

For simplicity, let us assume all Qi are positive, r ≥ 0 and k ≥ 0. After quotienting by

U(1), the φi configuration space is a weighted projective space with size determined by

the right-hand side of (3.2). This space is non-compact since Re(y) is only bounded from

above,

Re(y) ≤ r

2k. (3.3)

When the inequality in (3.3) is saturated, the weighted projective space collapses to zero

size. Although no fields charged under the gauge symmetry have expectation values at this

boundary, Σ is still massive because of the Y Σ coupling which contributes a |σ|2 mass term

to the physical potential. There is therefore no classical Coulomb branch emitting from

|φi|2 = 0. Just like its dual Stueckelberg field P , the field Y gives a mass to the gauge field

everywhere, so in the limit e→∞ the gauge field is not dynamical.

The simplest example has n chiral fields Φi of charge 1 and one twisted chiral Y with

the coupling (3.1) and a standard kinetic term. After carrying out the symplectic quotient,

we obtain a metric describing CP n−1 parametrized by n− 1 complex coordinates zi fibered

over a cylinder parametrized by y,

ds2 = R(y)

(dz · dz1 + |z|2

− |z · dz|2

(1 + |z|2)2

)+

(1 +

k2

R(y)

)dydy, (3.4)

R(y) = r − 2kRe(y) (3.5)

and also a B-field

B =kz · dz ∧ dy + kz · dz ∧ dy

1 + |z|2. (3.6)

15

We can write y = a+ iθ, absorb r into a, and define ρ2 ≡ R = 2ka to rewrite the metric in

a different form

ds2 = ρ2

(dz · dz1 + |z|2

− |z · dz|2

(1 + |z|2)2

)+

(1 +

ρ2

k2

)dρ2 +

(1 +

k2

ρ2

)dθ2. (3.7)

This form makes it clear that ρ has range (0,∞), with the projective space pinching to zero

size at ρ = 0, while the circle parametrized by θ becomes infinitely large at that end. This

is the trumpet geometry of Figure 1. This geometry is singular with diverging curvature as

ρ tends to zero. For example, when n = 2, the Ricci scalar is

R =4

ρ2+ . . . as ρ→ 0. (3.8)

While the space is geometrically singular, the theory has no physical singularity; the reso-

lution of the singularity requires T-duality and we will be discussed in section 3.2.

We can also calculate

H = dB = 2kdθ ∧ JFS (3.9)

where JFS is the fundamental two-form of Fubini-Study for CP n−1, which integrates to a

non-trivial torsion ∫C×S1

H = 4πk (3.10)

where C is the two-cycle dual to JFS, where the dual is taken at fixed ρ in the CP n−1 fiber,

and the S1 is parametrized by θ.

In this model we can pick an R-symmetry transformation for Y such that n − kγ =

0, so that there is a non-anomalous U(1)A, and we expect a non-trivial infrared CFT.

Using (2.32), the central charge of this CFT is calculated to be

c = 3n

(1 +

2n

k2

). (3.11)

3.1.2 Two U(1) actions

With at least one more U(1) gauge-field, we can bound the range of Re(y). Introduce a

second FI-term which couples the same field Y to the new field strength Σ,

− k

8π

∫d2xdθ+dθ− Y Σ + c.c., (3.12)

16

with charged fields φ satisfying ∑i

Qi|φi|2 = r − 2kRe(y). (3.13)

As long as k ≤ 0, r ≥ 0, and the charges Qi, Qi > 0, the range of Re(y) is bounded:

r

2k≤ Re(Y ) ≤ r

2k. (3.14)

This gives the cylindrical fixture of Figure 2 with a product of weighted projective spaces

fibered over the cylinder. In this basic fixture, the size of each projective space vanishes at

one of the ends.

Once again, the simplest models have n fields Φi with charges (1, 0) and n fields Φı with

charges (0, 1). Taking k ≥ 0, k ≤ 0 as in (3.14), we find the metric and B field

ds2 = R(y)

(dz · dz1 + |z|2

− |z · dz|2

(1 + |z|2)2

)+ R(y)

(dz · d¯z

1 + |z|2− |¯z · dz|2

(1 + |z|2)2

)+

(1 +

k2

R(y)+

k2

R(y)

)dydy, (3.15)

B =kz · dz ∧ dy + kz · dz ∧ dy

1 + |z|2+k ¯z · dz ∧ dy + kz · d¯z ∧ dy

1 + |z|2. (3.16)

We can calculate the H flux from B,

H = 2dθ ∧(kJFS + kJFS

), (3.17)

which is easy to integrate over the two-cycle dual to either JFS or JFS, and the S1 formed

by θ, ∫C×S1

H = 4πk,

∫C×S1

H = 4πk. (3.18)

Note that while k and k must have opposite signs in order to bound Y , they do not

necessarily have the same magnitude, and the same is true of the corresponding H-fluxes.

This is somewhat surprising because we might have expected the total brane and anti-brane

charge to sum to zero for a compact space. However, this does not seem to be a requirement

for these geometries. The dual descriptions of models with k 6= −k, which will be discussed

in section 3.2.2, involve either squashed weighted projective spaces, or spaces with orbifold

singularities.

When each fibered projective space is actually a sphere, which happens for P1 ∼ S2, the

resulting space is S5 × S1 [17]. Otherwise the space looks singular and, as we will discuss

17

shortly, we will need the T-dual description to see that the collapsing projective space is

actually acceptable.

Near each end, the metric has the same asymptotic form as (3.7), with the extra pro-

jective space staying at finite size. The spaces discussed in this section so far have been

previously studied in a (0, 2) context in [17].

3.1.3 Compact and conformal models?

The model with two U(1) gauge fields gives us a construction of a torsional compact ge-

ometry. However, there is no U(1)R charge assignment for y that allows us to solve (2.18),

and so this is a massive model. We can show that compact models will be generically

massive if we only allow Φ and Y fields with couplings of the form Y Σ and usual chiral

superpotentials.

As we have seen in the previous sections, we obtain one bound on the range of Re(y)

from each D-term condition, as long as all fields Φi charged under the corresponding U(1)

have positive charges (or all negative charges). The bound is of the form∑iQi|φi|2∑j Qj

≥ 0⇒ r − 2kRe(y)∑j Qj

≥ 0⇒ 2kRe(y)∑j Qj

≤ r∑j Qj

. (3.19)

This allows us to see that the direction of this bound depends only on the sign of∑j Qj

k.

But that sign is precisely what sets the sign of γ satisfying (2.18). Therefore if we have two

bounds in opposite directions as needed to make the range of Re(y) compact, there is no γ

which solves (2.18) for both U(1), and we will be dealing with a massive model.

We may now illustrate this argument with an example of an attempt to evade it, in

order to provide some intuition on how compactness is violated. Since the effects of Y are

generally not enough to cancel the U(1)R anomalies from two U(1) gauge fields, we can try

to also add a negatively-charged field coupled in a superpotential. This helps with cancelling

the anomaly, as it usually does for example in the quintic. However, since the models we

have been considering have a point where the size of the ambient projective space vanishes,

we would be left with a non-compact direction where Y and the negatively-charged field

grow without bound. We can try to be smarter and add an extra U(1).

Building on the double trumpet model, take then three U(1) gauge fields and the fol-

lowing field content: n fields Φi with charges (1, 0, 0), n fields Φi with charges (0, 1, 0), a

field S with charges (−Q,−Q, 0) and a field A with charges (0, 0, Qa). Introduce a periodic

twisted chiral Y , coupled to the three twisted chirals Σ, Σ,Σa with coefficients (k, k, ka).

As before we will want kk < 0 in order to obtain a bound, so choose k < 0. The field S can

18

be used to write a gauge-invariant superpotential Sf(Φ)g(Φ), with f and g polynomials of

degrees Q and Q, respectively. The D-term constraints read

−Q|s|2 + |φ|2 = r − 2kRe(y), (3.20a)

−Q|s|2 + |φ|2 = r − 2kRe(y), (3.20b)

Qa|a|2 = ra − 2ka Re(y). (3.20c)

From this it is clear that at the previous extrema of Re(y), φ or φ become zero, liberating

s and y in a non-compact direction. One of these can be removed by setting Q = 0. Once

we have done that, our anomaly cancellation conditions (2.18) read

n− γk = 0, (3.21a)

n− Q− γk = 0, (3.21b)

Qa − γka = 0. (3.21c)

Now we would like to use the bound from the third U(1) to remove the second non-compact

direction from the region φ = 0. However, note (3.21a) sets γk = n > 0 ⇒ γ > 0. Using

(3.21c), this implies Qa/ka > 0. Then the constraint imposed on y from (3.20c) will have

the same sign as that from (3.20a), and therefore will not help when the second D-term

constraint disappears and S becomes non-zero.

3.2 The Kahler picture

Both examples in sections 3.1.1 and 3.1.2 have T-duals which are Kahler and described by

the squashed toric varieties first discussed in [27]. At the level of an ultraviolet GLSM,

squashing is implemented as follows.

Consider a toric GLSM, namely a collection of n chiral superfields charged under a

collection of k abelian gauge symmetries. Such a model, in the absence of a superpotential,

has n − k remaining flavor symmetries. The squashing construction gauges each of these

flavor symmetries while simultaneously adding a Stueckelberg chiral superfield for each. An

19

action for such a model is

S =1

16π

∫d2xd4θ

(n∑i=1

k∑a=1

n−k∑α=1

|Φi|2e2Qai Va+2Rαi Vα −k∑a=1

2π

e2a

|Σa|2)

− i

8π

∫d2xdθ+dθ−

k∑a=1

taΣa + c.c (3.22)

+1

16π

∫d2xd4θ

n−k∑α=1

(bα2

(Pα + Pα + 2Vα

)2 − 2π

e2α

|Σα|2).

The charges of the chiral fields under the original k gauge symmetries are Qai , while the

charges of the flavor symmetries are Rαi . We stipulate that the combined n × n matrix

(Qai , R

αi ) has rank n. Note also that there are no FI couplings for the gauged flavor sym-

metries, as these can be absorbed into a redefinition of the corresponding Stueckelberg

fields.

Each squashing has an associated squashing parameter bα ∈ R. In the limit bα → ∞,

the Stueckelberg fields decouple, and the squashing is removed. Since each squashing

corresponds to a U(1) isometry, the Stueckelberg fields are periodic: ImPα ∼ ImPα + 2π.

Modifying the charge of Pα to kα for some kα ∈ Z yields a Zkα orbifold [27]. Note that in

section 3.1 we took bα = 1 for all fields Y . Taking the limit of no squashing in the Y -picture

leads to a Y field that has a vanishing classical kinetic term but gains a metric when we

descend to the non-linear sigma model. The torsion of the Y model is unaffected by the

value of the squashing parameter.

Applying the T-duality of section 2.2 results in the field-dependent FI couplings we have

already seen. The T-dual of (3.22) is

S =1

16π

∫d2xd4θ

(N∑i=1

k∑a=1

N−k∑α=1

|Φi|2e2Qai Va+2Rαi Vα −k∑a=1

2π

e2a

|Σa|2)

− i

8π

∫d2xdθ+dθ−

k∑a=1

taΣa + c.c (3.23)

− 1

16π

∫d2xd4θ

N−k∑α=1

(YαYαbα

+2π

e2α

|Σα|2)

− 1

8π

∫d2xdθ+dθ−

N−k∑α=1

YαΣα + c.c.

Dualizing a field Pα with shift-charge kα would lead to a factor kα multiplying the last line,

just like the couplings k in section 3.1. We will also mostly take bα = 1 to connect to the

previous discussion.

20

3.2.1 One U(1) action

The dual of the trumpet geometry discussed in section 3.1.1 has n chiral fields of charge 1

and a Stueckelberg field of charge k and period 2π with kinetic action

1

32π

∫d2xd4θ

(P + P + 2kV

)2.

This model was discussed in [21]. The total space is topologically Cn/Zk, but it is to easier

to visualize as S2n−1 warped over a half line R+. The orbifold action is through discrete

translations along the fiber of the Hopf fibration U(1) ↪→ S2n−1 → CP n. Note that if

k = 1, this space is birationally equivalent to the total space of the tautological bundle over

CP n−1. The classical metric in an affine patch of the CP n−1 base is

ds2 = ρ2

(dz · dz1 + |z|2

− |z · dz|2

(1 + |z|2)2

)+

(1 +

ρ2

k2

)dρ2 +

ρ2(1 + ρ2

k2

) (dθk

+ AFS

)2

, (3.24)

where θ has period 2π and AFS in this patch is

AFS = − i2

z · dz − z · dz1 + |z|2

. (3.25)

This is, indeed, the T-dual of (3.7) using the B-field (3.6).

This type of T-duality also provides us with a recipe for T-dualizing in the UV GLSM

along a given circle isometry in a non-linear sigma model. Once we identify the GLSM

circle that descends to the circle we would like to dualize, we can gauge the flavor symmetry

corresponding to that isometry, and add a corresponding Stueckelberg field P . We can then

dualize P into a Y as in Appendix B, and descend to the non-linear sigma model in the

resulting theory. Finally, if we want to remove the effects of the new field, we should take

the limit of no squashing, or b → ∞. It would be interesting to try this prescription in

models with blowing-up circles, such as those of [16].

3.2.2 Two U(1) actions

This dual has n chirals of charge (1, 0) and n chirals of charge (0, 1) and a single Stueckelberg

with charge (k, k) and period 2π with kinetic term

1

32π

∫d2xd4θ

(P + P + 2kV + 2kV

)2

.

As before, we will assume that k > 0 and k < 0 in order that the target space be compact.

Let k = gcd(k,−k). We can perform an SL(2,Z) transformation on the gauge fields(V

V

)=

(k

k

k

k

−˜ `

)(V

V

), k`+ k ˜= k, (3.26)

21

U(1)1 U(1)2 U(1)

Φ 1 0 0

Φ 0 1 0

Φ 0 0 1

S 0 0 −5

Table 1: Charge matrix for the quintic fibration over the double trumpet.

yielding a theory with n chirals of charge (`,− k

k) and n chirals of charge (˜, k

k) and a

Stueckelberg field of charge (k, 0) and period 2π with kinetic term

1

32π

∫d2xd4θ

(P + P + 2kV

)2

.

Matching onto the general model, we see this describes a weighted CP n+n−1 with n weights

of − k

kand n weights of k

k. Recall that all these weights are positive because k < 0. Further,

this space is squashed and orbifolded along the U(1) isometry under which the n chirals

have charge ` and the n chirals have charge ˜.

3.3 More general fibrations

The models described so far have both a Kahler and a torsional description, related by

duality. We would like to find models which do not have a Kahler description, and hence

live beyond the lamp post.

There is a very natural way to construct such models. Imagine a base GLSM theory

with Y -fields and some charged Φ fields. We will fiber a twisted sigma model over this

base theory in a way that obstructs dualizing Y back to a chiral superfield. We will fiber

the complex structure of the twisted sigma model over Y using superpotential couplings

between Y and charged twisted chirals Φ.

As a first example modeled on the quintic, consider a theory with a U(1)2 gauge group

with charged chirals and a U(1) gauge group with charged twisted chirals. The charge

matrix is given in Table 1.

We will not worry about imposing conformal invariance for the moment. Rather, our

interest is in the structure of the resulting generalized Kahler geometries. The chiral content

is the same as in section 3.1.2, and so the range of Re(y) will be bounded. We do not need

to assign any R-symmetry transformation to Y . On the other hand, the twisted chiral

22

content is similar to the usual quintic, except that we allow the twisted superpotential to

depend on eY :

W = Sf(eY , Φ), (3.27)

where f is a polynomial of degree 5 in Φ whose coefficients are polynomials in eY .

For concreteness, we can consider deforming the Fermat quintic by a Y -dependent mono-

mial, so

f(eY , Φ) =5∑i=1

Φ5i + eY Φ1Φ2Φ3Φ4Φ5. (3.28)

This Y -dependence cannot be removed by a field redefinition; the complex structure modu-

lus parametrized by Φ1Φ2Φ3Φ4Φ5 is now fibered over the Y cylinder. This space is therefore

a fibration of the quintic CY 3-fold over the double trumpet model. For a suitable choice of

parameters, the complex structure of the fiber can be kept away from degeneration limits.

This is the structure illustrated in Figure 3.

It is easy to generalize this structure to more general fibrations and bases resulting

in compact intrinsically torsional spaces. If, however, one wishes to impose conformal

invariance on the resulting space then we encounter the flat direction issue described in

section 3.1.3. The other natural possibility is to include eY couplings to Σ fields. That case

is rather interesting and will be discussed in section 3.4.

3.3.1 A comment on squashed Calabi-Yau

Although we run into the flat direction issue when trying to build compact conformal models

from Y fixtures, we can certainly repeat the usual hypersurface or complete intersection

construction of compact Calabi-Yau spaces for squashed projective spaces in terms of P

variables.

Let us describe these models by way of an example: a squashed analogue of the quin-

tic Calabi-Yau three-fold. Specifically, the model will describe a hypersurface inside of a

squashed CP 4 that is topologically Calabi-Yau. Recall the squashed CP 4: start with a

model with 5 chiral fields Φi, each with charge 1 under a single U(1)G gauge symmetry.

We choose to squash the flavor symmetry U(1)F under which only Φ5 rotates with charge

1. To do this, we gauge this symmetry and add a chiral Stueckelberg field P with kinetic

termb

32π

∫d4θ

(P + P + 2VF

)2,

where VF is the vector superfield of the gauged flavor symmetry. As usual, we choose ImP

to have period 2π. In order that the squashed CP 4 be smooth, we choose the shift charge of

23

U(1)G U(1)F

Φ1−4 1 0

Φ5 1 1

S −5 0

P 0 1

Table 2: Charge matrix for the squashed quintic model.

P to equal one; we can effect an orbifold of the ambient space by choosing another integer

charge `.

To carve out a hypersurface, we include another chiral field S with charge −5 under

the original gauge symmetry and charge 0 under the flavor symmetry, and we add the

superpotential

W (S,Φi, eP ) = S

(G(Φ1, . . . ,Φ4) + e−5PΦ5

5

).

The polynomial G is homogeneous of degree 5 and non-singular in the subspace Φ5 = 0. A

summary of the fields and their charges is provided in Table 2.

This model is of the form described in (2.22), and we can build a protected supercon-

formal multiplet in Q+-cohomology with the following parameter choices

γ = −β = 1, α5 = −1, αi = 0, i = 1, . . . , 4, αS = 1. (3.29)

As usual, this choice is ambiguous up to shifts by the charges under the original U(1) gauge

action. Now we expect this construction to give the correct central charge for the non-

compact total space ofO(−5) over the squashed CP 4, prior to turning on the superpotential:

cnon−cpt = 9 +6

b. (3.30)

This central charge depends on the squashing parameter b > 0. However, there is an

interesting question of the correct IR description of this theory with the superpotential

turned on. By a field redefinition φ5 → e−pφ5, the effect of the squashing can be removed

from the F-term constraints. The only effect of the squashing is a change of D-terms.

However, the IR theory is expected to be fully determined by the F-term structure if the

metric is compact, so our expectation is that this theory flows to the usual quintic Calabi-

Yau conformal field theory with c = 9 in the IR.2

2We would like to thank Ilarion Melnikov for clarifying this issue.

24

What we do learn from this construction is that there should be a corresponding relevant

operator in Y variables that also produces a compact CFT. The Y description contains two

U(1) gauge-fields with D-term constraints,

|φ1|2 + |φ2|2 + |φ3|2 + |φ4|2 + |φ5|2 − 5|S|2 = r1, (3.31)

|φ5|2 = r2 − 2Re(y). (3.32)

A part of that relevant operator is just the superpotential couplings involving fields that

are not dualized:

W (S,Φi, eP ) = S (G(Φ1, . . . ,Φ4)) .

This interaction still leaves flat directions for the physical potential. To lift those remaining

flat directions, we need the dual of the chiral operator e−5PΦ55. However, as is often found

in dual descriptions, this nice local chiral operator has no simple local description in Y

variables.

3.4 Exponential couplings

We now consider more exotic models that also obstruct a straightforward dualization to

a Kahler picture. These models break the U(1) symmetry shifting Im(Y ). That circle is

precisely the one we would want to T-dualize to produce a Kahler picture in terms of P

variables.

3.4.1 One U(1) action

As a first example of such a model, consider a theory with one Y , one U(1) gauge-field and

a twisted superpotential

S = − 1

8π

∫d2xdθ+dθ−κeY Σ. (3.33)

This model is the analogue of Figure 1, and the new coupling expressed in component

fields is found in (A.15). This coupling explicitly breaks the U(1) isometry which shifts the

imaginary part of Y . It should also be noted that while the coupling k in previous sections

was integer, κ can take any real value. This coupling also fixes Y to be invariant under

R-symmetry transformations, so Y can no longer be used to absorb any possible anomalies.

Writing y = a+ iθ, the D-term potential condition now reads∑i

Qi|φi|2 = r − 2κea cos θ = R(y), (3.34)

25

so if all the charges Qi are positive the space is bounded to the region

2κea cos θ ≤ r. (3.35)

Topologically, the space of possible y values satisfying this inequality can have one of three

shapes, depending only on r. These three possibilities are depicted in Figure 4.

• If r = 0, the condition (3.35) picks out one sign of the cosine for any a, and we obtain

an infinite strip.

• If r < 0, the condition imposes a bound on a, and we find a semi-infinite strip.

• If r > 0, on the other hand, we essentially obtain the converse of that, an infinite

cylinder with a semi-infinite strip taken out.

Figure 4: The three possibilities.

The metric and B-field can be obtained using similar methods from the above examples,

and have the form

ds2 = R(y)

(dz · dz1 + |z|2

− |z · dz|2

(1 + |z|2)2

)+

(1 +

κ2e2a

R(y)

)dydy, (3.36)

B =κeyz · dz ∧ dy + κeyz · dz ∧ dy

1 + |z|2. (3.37)

As above, the metric blows up at the boundary R(y) = 0. Define

ρ2 = r − 2κea cos θ, (3.38)

α = −κea sin θ, (3.39)

so

dydy = da2 + dθ2 =ρ2dρ2 + dα2

κ2e2a, (3.40)

and the metric becomes, in the limit close to the boundary,

ds2 = R(y)ds2FS + ρ2ds

2

FS + dρ2 +1

ρ2dα2, (3.41)

26

which has a similar form to the metric close to the boundaries in trumpet models, as can

be seen by comparison with (3.7).

The torsion in this case can be obtained from (3.37) and has the form

H = 2κd(ea sin θ) ∧ JFS = −2dα ∧ JFS (3.42)

This three-form integrates to zero in any region where the θ circle closes, but has a non-zero

integral in other regions.

3.4.2 Several U(1) actions

In order to build a compact space including an exponential coupling to a field strength

multiplet, start by taking the field content of section 3.1.2 which led to Figure 2. This

theory includes two U(1) gauge fields Σ and Σ coupled to two sets of chiral fields Φi and

Φi, leading to the D-term conditions:∑i

Qi|φi|2 = r − 2kRe(y) = R(y), (3.43)∑i

Qi|φi|2 = r − 2kRe(y) = R(y). (3.44)

This bounds the range of Re(y) if all charges Qi and Qi are positive, k > 0, and k < 0. To

this configuration, which is dual to a squashed space, we can now add a third U(1) multiplet

Σ′ with its own set of charged fields Φ′, coupled to Y with a superpotential κeY Σ′. The

corresponding D-term condition reads∑i

Q′i|φ′i|2 = r′ − 2κea cos θ = R′(y). (3.45)

The boundaries of the space are set by the y values where any single projective space

collapses to zero size. We encounter a problem with new flat directions if any two projective

spaces collapse to zero size at the same y value. To see this, note that the Y superpotential

can only mass up a single combination of Σ fields. If two or more projective spaces collapse

at the same point, there will be two distinct U(1) factors for which no charged fields have

an expectation value at that point. Therefore, one Σ multiplet will be massless resulting

in a new flat direction.

It is straightforward to see that to avoid an intersection where two or more projective

spaces collapse, we need to be in the case where the exponential allows the full y circle,

so r′ > 0. This is the last case depicted in Figure 4. We also need the boundary for the

27

projective space with the exponential coupling to be fully outside the space defined by the

other two constraints. The set of a satisfying the condition

ea ≥ r′

2|κ|, (3.46)

contains a boundary point for some θ where the projective space with the exponential

coupling vanishes. We therefore need to impose the condition

logr′

2|κ|>

r

2k, (3.47)

to ensure these boundary points are excluded. We will define the theory at a UV scale Λ

with bare FI parameters satisfying inequality (3.47). The inequality will then generally be

preserved by RG flow because the additive renormalization of the FI parameters makes the

right-hand side decrease faster than the left-hand side as we flow down in energy.

The metric and B-field are essentially the sum of those given in sections 3.1.2 and

3.4.1. This model then consists of the same ingredients as those of the double trumpet

model, with an additional space fibered over the double trumpet whose size depends on

the real and imaginary values of y. This fibration structure is depicted in Figure 5. The

integrated torsion will have the same value as found in the double trumpet. However, since

the isometry is broken by the exponential coupling, this model cannot be dualized into a

Kahler picture like a squashed projective space.

3.5 Unifying constructions

We can unify the structures described in sections 3.1 and 3.4 by writing the Σa coupling as

fa(Y )Σa, with an fa(Y ) that shifts at most by an integer multiple of 2πi when Y shifts by

2πi. For simplicity, take each φ to have charge 1 under one of the gauge symmetries, and

0 under the others. Then the D-terms constrain∑i

|Φai |2e2Aa = ra − 2 Re fa(y) = Ra(y), (3.48)

and repeating the analysis yields a metric of the form

ds2 =∑a

Ra(y)ds2(φa) +∑µ

|dyµ|2 +∑a

|dfa|2

Ra(y), (3.49)

Near Ra = 0 it makes sense to choose coordinates including ρ2 = fa, and in that limit the

ρ metric will reduce to the familiar form (3.7). The B field can also be written in terms of

28

Re(y)

Im(y)

Figure 5: A compact example with exponential couplings. The three fibrations are depicted.

The first two fibrations are the ones already seen in Figure 2. The radius of the third space

depends on both the real and imaginary parts of y. Note that the radius oscillations become

larger as Re(y) becomes bigger.

the functions fa as

B =∑a

φa · dφa ∧ dfa1 + |φa|2

+ c.c. (3.50)

⇒ H =∑a

2d (Im fa) ∧ JaFS. (3.51)

If the circle from the imaginary part of y closes, we can see from this expression that the

integral of H will be non-zero if fa is not single-valued when we travel around the circle.

29

4 The Quantum Cohomology Ring

Our discussion so far has been largely classical. We wanted to describe gauge theories that

give rise to classical vacuum equations describing generalized Kahler spaces. In this section,

we turn to some quantum aspects of these models. Specifically, we will probe the vacuum

structure by calculating the quantum cohomology rings for some of the models we have

discussed.

4.1 Coulomb branch vacua

In (2, 2) gauge theories, we can investigate vacua where the scalars in field strength mul-

tiplets Σa gain expectation values. In this section, we review the basic ingredients needed

to describe these Coulomb branch vacua. We can see from (A.9) that all fields φi charged

under the gauge symmetries become massive when each Σa has an expectation value. Such

fields can then be integrated out at one loop, leading to a quantum correction to the twisted

superpotential. The effective superpotential has the form [11,35,36]

SW =1

8π

∫d2xdθ+dθ−

∑a

Σa

(∑i

Qai

[log

(∑bQ

biΣb

Λ

)− 1

]− ita

)

=1

8π

∫d2xdθ+dθ−

∑a

Σa

(∑i

Qai

[log

(∑bQ

biΣb

µ

)− 1

]− ita(µ)

)(4.1)

where Λ is the UV renormalization scale, ta are the bare FI parameters, and ta(µ) include

the renormalization

ta(µ) = ta(Λ) + i

(∑i

Qai

)log

µ

Λ. (4.2)

Note that fields P that are shift-charged under the gauge symmetries or fields Y with a Y Σ

superpotential are not massed up in the same way. This expression is only valid at large

values of Σa, where the masses of the fields that have been integrated out are large, so once

a solution is obtained it should be checked that we are in the right regime. Varying this

superpotential leads to the vacuum equations

∑i

Qai log

(∑bQ

biΣb

µ

)= ita(µ)⇒

∏i

(∑bQ

biΣb

µ

)Qai= eita(µ). (4.3)

Taking the exponential of both sides here does not alter the solutions to the equation,

since the right-hand side includes iθ, with its 2πi periodicity. Since we will be discussing

30

generalizations of this structure, we note that supersymmetric vacua on the Coulomb branch

are determined by solutions to

exp

(∂Weff

∂Σa

)= 1 (4.4)

for all field strength multiplets Σa, where Weff is the effective twisted chiral superpotential

obtained by integrating out all the charged fields. If there are additional twisted chiral fields

which are not field strength multiplets, like Y fields, then we also impose the condition

∂Weff

∂Y= 0 (4.5)

for each such field Y .

As a warm-up for the models considered in this work, we can use this superpotential to

find the vacuum structure of the CP n−1 gauged linear sigma model. The field content of

this model consists of one Σ and n chiral fields Φi of charge 1. A straightforward application

of the formulae above leads to the superpotential

SW =1

8π

∫d2xdθ+dθ−Σ

(n

[log

(Σ

µ

)− 1

]− it(µ)

), (4.6)

giving n critical points obeying

Σn = µneit(µ). (4.7)

As we move to lower energies, we can see from (4.2) that Re(it(µ)) = −r grows large. This

means that the mass of the fields Φi that were integrated out, given by |Σ|, grows relative

to µ, and therefore the calculation is justified.

4.2 Double trumpet

We now analyze the simplest model we presented with a bounded range for y, the double

trumpet model introduced in section 3.1.2. To study the Coulomb branch, we take Σ, Σ to

have non-zero vacuum expectation values. This causes the fields Φ, Φ to become massive,

and we can integrate them out. Note that Y is not massed up by these expectation values,

since there is no yσ potential. The effective superpotential was computed in section 4.1.

The equations determining supersymmetric vacua take the form:

kΣ + kΣ = 0, (4.8a)

n log

(Σ

µ

)− kY = it, (4.8b)

n log

(Σ

µ

)− kY = it, (4.8c)

31

These three equations can be solved for Y,Σ and Σ, leading to a set of ring relations between

these twisted chiral operators.

There are some basic issues to understand in this Y picture. The first issue is one of

identifying observables. In conventional Kahler GLSM models, each Σa field is associated to

an FI parameter and therefore to a Kahler class. For non-Kahler models, the map between

Coulomb branch operators and observables on the Higgs branch is not a priori clear. For

this particular model, each field Σ, Σ and Y is a twisted chiral superfield with a bottom

component that is (Q+ +Q−)-closed, but not exact. The Y field is distinguished from the

Σ, Σ fields because it is not a field strength multiplet. Here duality helps us determine the

observables because we know this model is equivalent to a Kahler model with a P field. In

that picture Σ, Σ are possible observables but not Y . Eliminating Y from (4.8b) and (4.8c)

gives the relation:

− kn log

(Σ

µ

)+ kn log

(Σ

µ

)= −ikt+ ikt ⇒ Σ−knΣkn = µ−kn+kne−ikt+ikt. (4.9)

We can then use (4.8a) to express this relation in terms of a single Σ,(−kk

)knΣ−kn+kn = µ−kn+kne−ikt+ikt. (4.10)

This is the quantum cohomology ring for the double trumpet model.

We can ask whether this ring is fundamentally different from the ring one would find

in a toric case with no Y couplings. While the basis of gauge symmetries we used for this

model is the most convenient to see compactness of the Y interval, it is not the best basis

to understand the ring. We saw in section 3.2.2 that we can change the basis of gauge fields

for this model in order to transform it into the dual of a squashed toric model with n fields

of charge −k/k and n fields of charge k/k, where k = gcd(k,−k). Note all these charges

are positive.

In this basis, Y is only coupled to one field strength multiplet, which is precisely the

combination of the original gauge field multiplets appearing in (4.8a). That multiplet will

be set to zero by the Y equation of motion. The other field strength multiplet will obey a

condition that is a function only of its FI parameter and the charges of the fields integrated

out. The ring we constructed above in (4.10) is therefore the quantum cohomology ring of

a weighted projective space.

We can apply the same reasoning to any model which is either a squashed Kahler model

or a dual description of a squashed Kahler model. This is the case because squashing

32

is a modification which only affects the D-terms, but not the F -terms which determine

the quantum cohomology ring. We can therefore turn squashing off without changing the

resulting ring.

4.3 Exponential models

The model in section 3.4.1 does not have Coulomb branch vacua because the eY Σ coupling

with only one U(1) factor always sets Σ = 0.

The Coulomb branch vacuum structure for the model of section 3.4.2 is determined from

the critical points of the effective twisted chiral superpotential:

SW =1

8π

∫d2xdθ+dθ−

[Σ

(n log

(Σ

µ

)− n− kY − it(µ)

)+ Σ

(n log

(Σ

µ

)− n− kY − it(µ)

)

+ Σ′(n′ log

(Σ′

µ

)− n′ − κeY − it′(µ)

)]. (4.11)

These critical points satisfy the equations,

kΣ + kΣ + κeY Σ′ = 0, (4.12a)

n log

(Σ

µ

)− kY = it(µ), (4.12b)

n log

(Σ

µ

)− kY = it(µ), (4.12c)

n′ log

(Σ′

µ

)− κeY = it′(µ). (4.12d)

Solving the three latter equations leads to Σ solutions for any value of Y ,(Σ

µ

)n= eit(µ)+kY ,

(Σ

µ

)n

= eit(µ)+kY ,

(Σ′

µ

)n′= eit

′(µ)+κeY . (4.13)

Each of these equations has a finite number of solutions for a fixed Y , giving a total of nnn′

vacua.

Just like the case considered in (4.7), it is valid to integrate out the charged fields as long

as the right-hand sides of the equations appearing in (4.13) are large. In equation (4.12b),

the FI parameter t will run according to (4.2),

it(µ) = it(Λ)− n logµ

Λ, (4.14)

33

and therefore Σ/µ will scale with 1/µ as µ becomes smaller. Equivalently, solutions for Σ

will be independent of µ. The n-dependence dropped out of this argument; it also drops

out of analogous arguments which apply to equations (4.12c) and (4.12d). This implies

that once we plug in Σ into (4.12a), it can also be solved for Y independently of µ. The

masses of the fields we integrated out will therefore become large when compared to µ for

all vacuum solutions by the same argument found in section 4.1.

We still have to solve the first equation, (4.12a). To count the number of solutions, it is

easier to work with the single-valued field X = eY . In terms of X, the remaining vacuum

equation takes the form:

keit/nXk/n + keit/nX k/n + κXeit′/n′+κX/n′ = 0. (4.15)

This equation has no dependence on the scale µ. This can be seen by noting that µ drops

out of equations (4.12) and (4.13). So one can use t, t and t′ defined at the scale Λ in

equation (4.15). As a complex function of X, the left-hand side of this equation has an

infinite number of zeroes, so we have infinite distinct vacua in the Coulomb branch. Since

the masses of the integrated out fields are large in the IR, there is no obvious problem with

this analysis.

The structure we have found here is quite surprising and quite different from usual

computations of quantum cohomology. It might well be indicative of a more generic vacuum

structure found when examining generalized Kahler spaces beyond the lamp post. There

are a couple of points to summarize: first, the Higgs branch geometry is compact for this

model. In fact, the condition for the space to remain compact found in (3.47) can easily be

preserved under RG flow. However, the space is non-Kahler and the structure of instanton

corrections, which usually generate quantum cohomology, has yet to be understood in any

detail. Similar comments apply to the observables of the theory. Because there is a non-zero

H, the instanton configurations are likely to be complex field configurations.

What we see is an infinite number of Coulomb branch vacua for this model. It is possible

that further quantum corrections will lift these vacua, but since these vacua are seen from

a holomorphic superpotential, it is not clear from where these quantum corrections might

originate. One possibility are strong interactions between Y and Σ generating an anomalous

dimension for operators like eY . If we want to interpret the Coulomb branch as an operator

ring, capturing quantum corrections to a classical ring of observables associated to the Higgs

branch, then we note that X = eY must be retained as an operator. The four operators

(Σ, Σ,Σ′, X) then satisfy the ring relations (4.13) and (4.15).

One other possibility is that the Coulomb branch of this model should not be viewed

34

U(1) U(1)s U(1)′

Φi (×n) 1 0 0

Φs (×1) 1 1 0

Φ′j (×n′) 0 0 1

Table 3: The charge matrix for the example of section 4.4.

as purely encoding data interpretable in terms of Higgs branch physics. In conventional

GLSM examples, Σ fields can be related to Higgs branch fields via equations of motion.

In this case, this is still true for the Σ fields but the neutral Y field does appear on both

branches, which perhaps suggests that the Coulomb branch might be viewed as distinct

from the Higgs branch.

4.4 An explicit example

Let us remove some of the notational clutter to better understand and interpret what

might be going on. Take the case k = −k = 1. Additionally, take n = 1. We also want a

better feel for how the exponential coupling with coefficient κ is so dramatically changing

the vacuum structure. So we will change basis for (Σ, Σ), as outlined in section 4.2, to

(Σ,Σs) = (Σ, Σ− Σ). This is a basis in which the U(1) factors transparently describe the

dual of a squashed CP n model. We have n fields Φi with charge 1 under the first U(1)

factor Σ, which has no Y coupling; there is one field Φs with charge 1 under Σ and also

charge 1 under Σs. The Σs gauge symmetry has a Y coupling. The phase of the Φs field

is squashed in the P picture. Finally there are fields Φ′j with charge 1 under Σ′. This

collection of fields appears in table 3.

Supersymmetric vacua are determined by equations (4.12) which become

−Σs + κeY Σ′ = 0, (4.16a)

n log

(Σ

µ

)+ log

(Σ + Σs

µ

)= it+ it, (4.16b)

log

(Σ + Σs

µ

)+ Y = it, (4.16c)

n′ log

(Σ′

µ

)− κeY = it′. (4.16d)

The first thing we would like to recover is the ring of the dual projective space, which

35

should emerge in the limit κ→ 0. Setting κ = 0 forces Σs = 0 from (4.16a). The remaining

equations then decouple and in terms of X = eY we find:

Σn+1 = µn+1ei(t+t), Σ′n′= µn

′eit′, ΣX = µeit. (4.17)

The first two relations are the familiar ones we expect for the CP n × CP n′−1 model. The

last relation constrains the operator X in terms of Σ.

Now we turn on κ. It still seems natural to use (4.16a) to solve for Σs with Σs = κXΣ′.

The rest of the equations give the relations,

Σn(Σ + κXΣ′) = µn+1ei(t+t), Σ′n′e−κX = µn

′eit′, X (Σ + κXΣ′) = µeit, (4.18)

which are an intriguing deformation of the ring relations (4.17). Specifically, Σ and Σ′ are

now coupled as we might expect from figure 5. This appears to be the case even in the

classical limit where r, r, r′ →∞, where the ring should correspond to a geometric ring of

the generalized Kahler space.

Once we take κ 6= 0, the number of solutions to the equation determining X,

eit/nX1/n − eitX−1 + κXeit′/n′+κX/n′ = 0, (4.19)

moves from finite to infinite. It is worth noting that if we try to perturbatively expand the

zero solutions around κ = 0 to any finite order in κ, the number of solutions will still be

finite. There are an infinite number of solutions that are not analytic around κ = 0. To

make this more natural, we can examine a toy model with only one chiral superfield X and

a superpotential of the form

W = X − eκX . (4.20)

Critical points of this superpotential obey the condition

1− κeκX = 0. (4.21)

When κ = 0, this equation has no solutions. However, if κ > 0, the solutions are given by

X =1

κlog

(1

κ

). (4.22)

There is an infinite set of solutions, one for each branch of the logarithm. They are all

non-analytic in κ, moving to infinite |X| as κ is taken to 0.

It would be very interesting to calculate elliptic genera for this class of models, in order

to better understand the infinity of vacua we have found. Unfortunately, to compute the

36

elliptic genus in a straightforward way, we need both U(1)L and U(1)R R-symmetries to be

unbroken, which is not true for these compact models. As we have already discussed, it is

challenging to find torsional examples which are both compact and conformal in this (2, 2)

setting. Such models are possible with (0, 2) worldsheet supersymmetry. For non-compact

models, the elliptic genus should generically have a non-holomorphic dependence on the

torus modular parameter; see, for example [37,38].

4.5 Unified structure

This Coulomb branch analysis can also be generalized to the unified case described in section

3.5. We allow for several fields Y µ, and any number of vector superfields Σa, coupled by

twisted superpotential couplings of the more general form fa(Y )Σa. We take the set of

charged fields to consist of na chiral superfields which are charged with charge 1 only under

one gauge symmetry corresponding to Σa. If we integrate out the charged superfields, we

obtain an effective twisted superpotential,

SW =1

8π

∫d2xdθ+dθ−

∑a

Σa

(na

[log

(Σa

µ

)− 1

]− fa(Y )− ita(µ)

). (4.23)

Varying the superpotential with respect to Σa gives equations of the form

na log

(Σa

µ

)− fa = ita ⇒

(Σa

µ

)na= eita+fa . (4.24)

Varying with respect to Y µ gives the conditions∑a

∂µfaΣa = 0. (4.25)

If we are interested in solving for vacua rather than studying rings, we can further substitute

solutions to (4.24) giving:∑a

∂µfaeita/na+fa/na = ∂µ

(∑a

naeita/na+fa/na

)= 0. (4.26)

As in the previous section, the masses of the fields that were integrated out become arbi-

trarily large as we flow to lower energies.

We now want to explore the generic number of solutions to (4.26). For simplicity, take

one Y field, and consider (4.26) initially as a complex function of the cylinder variable y.

For compactness a > 1. We want to characterize the number of zeros of this function,

h(y) =∑a

∂yfaeita/na+fa/na . (4.27)

37

The functions fa take the form

fa = kay + fa, fa =∞∑

m=−∞

cma emy,

for complex constants cma . Some ka might be negative. We would like to move from the

cylinder variable y ∼ y + 2πi to a single-valued variable. If it were not for the na factors

appearing in (4.27), we would simply use x = ey. Instead define nlcm = lcm{na}. We can

then view h(y) as a complex function of z = ey/nlcm rather than y. The penalty for this

change of variable is that the equation h(y) = 0 is replaced by a finite collection of equations

in z obtained by repeatedly shifting y → y + 2πi.

To proceed, we need to be able to say something about fa. Usually, we do not want

singular couplings in the classical Lagrangian so let us assume that fa is smooth with no

singularities for finite values of y. Viewed as a function of z, this implies fa is holomorphic

in z away from 0 and ∞. It is not particularly strange to also insist that fa is holomorphic

in z. At least under this restriction, we can say something more about the number of zeros

because h(z) is analytic in the complex plane. Any analytic function with a finite number

of zeros can be written in the form P (z)eg(z) where g(z) is also an analytic function. Our

h(z) takes the form,

h(z) =∑a

(ka + ∂fa(z)

)zkanlcmna e

fa(z)na e

itana , fa =

∞∑m=0

cma zm·nlcm . (4.28)

If all fa(z) are identical then h(z) can admit a finite number of zero solutions. Otherwise,

we generically expect an infinite number of solutions as we saw in the example of section 4.4.

Acknowledgements

It is our pleasure to thank L. Anderson, J. Gray, J. Halverson, C. Long, and W. Taylor

for discussions about the degeneracy and fibration structure of currently known Calabi-Yau

constructions. We would like to thank M. Dine, D. Kutasov, M. R. Plesser, and E. Sharpe

for helpful discussions. We are particularly grateful to I. Melnikov for very helpful comments

about these constructions. J. C. and S. S. are supported in part by NSF Grant No. PHY-

1720480.

38

A (2, 2) Superspace Conventions

A.1 Basic conventions

In this Appendix, we state our conventions. We use superspace coordinates,

(x+, x−, θ±, θ±),

where the metric is Lorentzian, x± = 12(x0 ± x1) and ∂± = ∂0 ± ∂1 so that ∂±x

± =

1, ∂±x∓ = 0. The superspace integration measure d4θ = dθ+dθ+dθ−dθ− is defined so

that∫d4θ θ−θ−θ+θ+ = 1. The Levi-Civita tensor is defined by ε01 = 1 so ε−+ = 1

2.

The supersymmetry charges and super-derivatives can be written in terms of superspace

coordinates as follows:

Q± = ∂θ± + iθ±∂±, Q± = −∂θ± − iθ±∂±, (A.1)

D± = ∂θ± − iθ±∂±, D± = −∂θ± + iθ±∂±. (A.2)

These operators satisfy the algebras

{Q±, Q±} = −2i∂±, {D±, D±} = 2i∂±. (A.3)

A (2, 2) GLSM is constructed from a collection of constrained superfields. The first

ingredient we use in this work are chiral superfields, Φ, which satisfy D±Φ = 0. They

contain the following component fields:

Φ = φ+√

2θ+ψ+ +√

2θ−ψ− + 2θ−θ+F + . . . , (A.4)

with all other terms involving derivatives of these fields. We also use twisted chiral super-

fields, Y , satisfying the conditions D+Y = D−Y = 0. The θ-expansion of these superfields

takes the form:

Y = y +√

2θ+χ+ +√

2θ−χ− + 2θ−θ+G+ . . . . (A.5)

To gauge an abelian global symmetry acting on chiral superfields, we introduce a U(1)

vector superfield V , which is a real superfield. In Wess-Zumino gauge, the vector superfield

has the expansion:

V = θ+θ+A+ + θ−θ−A− − θ−θ+σ + θ−θ+σ +√

2θ−θ+θ+λ+ −√

2θ−θ+θ+λ+

+√

2θ−θ−θ+λ− −√

2θ−θ−θ+λ− + 2θ−θ−θ+θ+D. (A.6)

39

A gauge transformation acts by sending,

V → V +i

2(Λ− Λ), Φ→ eiQΛΦ, (A.7)

where the chiral superfield Φ has chargeQ. From V we can build the field strength superfield

Σ = D+D−V = σ +√

2θ+λ+ +√

2θ−λ− + θ−θ+(−2D + iF−+) + . . . , (A.8)

which is gauge-invariant and twisted chiral by construction. Similarly, if we want to gauge

a global symmetry acting on twisted chiral superfields, we need to introduce a chiral vector

superfield V .

Armed with these ingredients, we can describe the basic couplings of a (2, 2) GLSM.

The canonical kinetic term of a chiral field Φ is given by

S =1

16π

∫d2xd4θ Φe2QV Φ,

=1

4π

∫d2x[− |Dµφ|2 + iψ+D−ψ+ + iψ−D+ψ− + |F |2 +QD|φ|2 −Q2|σ|2|φ|2 (A.9)

+Qλ−φψ+ +Qψ+φλ− +Qλ+φψ− +Qψ−φλ+ +Qψ+σψ− +Qψ−σψ+

].

Similarly canonical kinetic terms for a neutral twisted chiral superfield are given by

S = − 1

16πb

∫d2xd4θ Y Y,

=1

4πb

∫d2x

[−|∂µy|2 + iχ+∂−χ+ + iχ−∂+χ− + |G|2

], (A.10)

and finally the kinetic terms for a shift-charged chiral superfield are

S =b

32π

∫d2xd4θ(P + P + 2QV )2,

=b

4π

∫d2x[− |Dµp|2 + iψ+∂−ψ+ + iψ−∂+ψ− + |F |2 +QD(p+ p)−Q2|σ|2

+Qλ−ψ+ +Qψ+λ− +Qλ+ψ− +Qψ−λ+

]. (A.11)

Kinetic terms for the gauge-field are built from Σ,

S = − 1

8e2

∫d2xd4θ ΣΣ,

=1

2e2

∫d2x

[−|∂µσ|2 + iλ−∂+λ− + iλ+∂−λ+ +D2 − 1

2FµνF

µν

]. (A.12)

40

Superpotential couplings involve holomorphic combinations of chiral fields,

SW =1

8π

∫d2xdθ+dθ−W (Φ) + c.c.,

=1

4π

∫d2x

[∂iW (φ)F i + ∂ijWψi+ψ

j−]

+ c.c.. (A.13)

For holomorphic combinations of twisted chiral superfields, we can build an analogous

twisted chiral superpotential W . Specific examples central to our discussion involve the

gauge-field strength Σ and a Y field, taking the form

SW = − k

8π

∫d2xdθ+dθ− Y Σ + c.c.,

=k

4π

∫d2x [2 Re(y)D + Im(y)F−+ − (σG+ χ+λ− + λ+χ− + c.c.)] , (A.14)

or

SW = − κ

8π

∫d2xdθ+dθ− eY Σ + c.c., (A.15)

=κ

4π

∫d2x [2 Re(ey)D + Im(ey)F−+ − (ey(σG+ χ+λ− + λ+χ− + σχ+χ−) + c.c.)] .

Note that compatibility of these couplings with the periodicity Y ∼ Y + 2πi imposes the

constraint k ∈ Z, but there is no such restriction on κ. A more basic example is the

Fayet-Iliopoulos (FI) coupling for an abelian gauge-field, given by

SFI = − it

8π

∫d2xdθ+dθ−Σ + c.c. =

1

4π

∫d2x [−rD + θεµνFµν ] , (A.16)

where

t =ir

2+ θ. (A.17)

B Dualization

Here we summarize the various useful (2, 2) dual descriptions found, for example, in [18].

We will take special care to ensure the correct normalization for the Lagrange multiplier

terms so that circles in both the original and dual descriptions have 2π periodicity.

B.1 Component picture and periodicity

We will derive T-duality by proving equality of the path integrals over the two dual theories,

through the construction of a theory with auxiliary fields that can be transformed into either

41

of the dual theories. Schematically, we want to prove equivalence between a theory with

action

S = − b

4π

∫d2x∂µϕ∂

µϕ (B.1)

and one given by

S = − 1

4πb

∫d2x∂µθ∂

µθ, (B.2)

where both ϕ and θ are 2π-periodic real fields. The strategy will be to rewrite the first

theory replacing dϕ with an unconstrained one-form, adding the right Lagrange multiplier

so that the theories are equivalent.

On a topologically non-trivial worldsheet whose first cohomology group has rank n,

take {ωi}, i = 1, . . . , n to be a basis for the closed non-trivial one-forms dual to a basis of

non-trivial cycles such that the matrix∫ωi∧ωj = J ij is an element of SL(n,Z). A generic

closed one-form can then be written

c = cµdxµ = dϕ0 +

∑i

aiωi, (B.3)

with ϕ0 single-valued and ai real numbers.

To dualize ϕ in (B.1), we will substitute ∂µϕ→ cµ and add a Lagrange multiplier θ to

enforce dc = 0. The action then becomes

S0 = − b

4π

∫d2xcµc

µ + κ

∫θdc (B.4)

where κ is a multiplicative constant. When we integrate out θ, the equation dc = 0 allows

us to trivialize c = dϕ and return to the original theory. However, as in (B.3), ϕ in this

expression will not be a single-valued field if the worldsheet is topologically non-trivial, so

we need to find the relation between the periodicities of θ and ϕ induced by this coupling.

Defining the Lagrange multiplier θ to have period Tθ, we can expand dθ as

dθ = ∂µθdxµ = dθ0 + Tθ

∑i

niωi, (B.5)

with single-valued θ0 and integers ni. We then have∫c ∧ dθ =

∫cµ∂νθdx

µ ∧ dxν =

∫d2xεµνcµ∂νθ = Tθ

∑i,j

aiJijnj, (B.6)

where we added the intermediate forms for later use. Now when we perform the path

integral over θ, the θ0 part gives dc = 0, but the integral also includes a sum over ni, which

gives ∑nj

exp(−iκTθaiJ ijnj

)∝∏i

∑mi

δ(κTθai − 2πmi), (B.7)

42

constraining ai to be integer multiples of 2πκTθ

, so comparing to (B.3), we see that c = dϕ

where ϕ is periodic with period Tϕ = 2πκTθ

. Therefore, if we want Tθ = Tϕ = 2π, we should

take κ = 1/(2π).

Now that we have fixed the constant in (B.4), the dual action can be obtained by instead

integrating out cµ. Since the action is quadratic, the path integral sets cµ to the solution

of its classical equation of motion,

bcµ = −εµν∂νθ, (B.8)

or in components

bc± = ±∂±θ. (B.9)

The dual action we obtain is (B.2).

B.2 (2, 2) duality

Consider now a free (2, 2) chiral field P with periodicity P ∼ P + 2πi:

S =b

16π

∫d2xd4θ |P |2. (B.10)

To dualize the imaginary component of P , we will substitute P + P → 2B where B is a

generic real superfield. Denoting the imaginary part of P by ϕ and the one-form in B by

cµ, we see this corresponds to the component substitution in the previous section since

P + P = · · · − iθ+θ+∂+(p− p)− iθ−θ−∂−(p− p),

= · · ·+ 2θ+θ+∂+ϕ+ 2θ−θ−∂−ϕ, (B.11)

and we expand

B = · · ·+ θ+θ+c+ + θ−θ−c− (B.12)

so P + P → 2B is the (2, 2) extension of dϕ→ c.

Carrying out the substitution and including a Lagrange multiplier F , the action becomes

S0 =1

8π

∫d2xd4θ

[bB2 + FD+D−B − FD+D−B

],

=1

8π

∫d2xd4θ

[bB2 −D−D+FB + D−D+FB

],

=1

8π

∫d2xd4θ

[bB2 −B(Y + Y )

], (B.13)

43

where we integrated by parts and defined Y = D−D+F to obtain the second form of the

action. Note that Y is twisted chiral by definition.

If we integrate out the Lagrange multiplier F , its equation of motion is solved by setting

B to the sum of a chiral and its conjugate and we recover the original theory. We should

check that the factor multiplying the Lagrange multiplier term is the one determined in

section B.1: expand

Y + Y = · · ·+ 2θ+θ+∂+θ − 2θ−θ−∂−θ, (B.14)

from which we see

SL = − 1

8π

∫d2xd4θB(Y + Y )

= − 1

4π

∫d2x [c−∂+θ − c+∂−θ] + . . .

= − 1

2π

∫d2xεµνcµ∂νθ + . . . (B.15)

Therefore the periods of θ and ϕ will be related by TθTϕ = 4π2; if one period is 2π periodic,

so will the other.

Solving (B.13) for B instead, we find the dual action

Sd = − 1

32πb

∫d2xd4θ(Y + Y )2 = − 1

16πb

∫d2xd4θ|Y |2. (B.16)

This is a canonical kinetic term for a twisted chiral. Note the duality has mapped

b(P + P ) = Y + Y , (B.17)

which for their scalar component fields becomes

b(p+ p) = y + y, (B.18)

b∂±(p− p) = ±∂±(y − y). (B.19)

This confirms our assertion that we are performing a T-duality transformation on the

imaginary part of p. The imaginary part of p parametrizes a circle with radius√b, while

the circle parametrized by the imaginary part of y has radius 1/√b.

Such a field P can be axially charged, making its imaginary part a two-dimensional

Stueckelberg field. The action is simply

S =b

32π

∫d2xd4θ

(P + P + 2V

)2. (B.20)

44

Using the same substitution as above gives,

S0 =1

8π

∫d2xd4θ

[b(B + V )2 −B(Y + Y )

], (B.21)

which we can solve for B to find

Sd =1

8π

∫d2xd4θ

[− 1

4b(Y + Y )2 + (Y + Y )V

],

= − 1

16πb

∫d2xd4θY Y − 1

8π

∫d2xdθ+dθ−Y Σ + c.c.. (B.22)

The last term in the action is the coupling between Y and Σ in (2.6). Note that a field P

with charge Qp is dual to Y with coupling ky = Qp. The duality maps are similar to (B.19)

except the p side will now feature covariant derivatives D±p = ∂±p + A±, so both sides of

the maps are gauge-invariant.

Alternatively, if we start from a (2, 2) chiral Φ parametrizing a plane, we can redefine

Φ = eΠ to dualize the phase of Φ by similar techniques. Since (2, 2) theories are not chiral,

no non-trivial Jacobian results from this redefinition. Replacing Π + Π→ 2B gives,

S0 =1

16π

∫d2xd4θ

[e2B − 2B(Y + Y )

], (B.23)

from which we can once again solve for B to find

Sd = − 1

16π

∫d2xd4θ (Y + Y ) log(Y + Y ). (B.24)

The same procedure can be performed if Φ is charged,

S =1

16π

∫d2xd4θ |Φ|2e2V → S0 =

1

16π

∫d2xd4θ

[e2B+2V − 2B(Y + Y )

], (B.25)

which gives the dual action:

Sd = − 1

16π

∫d2xd4θ

[(Y + Y ) log(Y + Y )− 2(Y + Y )V

],

= − 1

16π

∫d2xd4θ (Y + Y ) log(Y + Y )− 1

8π

∫d2xdθ+dθ− Y Σ + c.c.. (B.26)

In this case, for the theories to be quantum-mechanically equivalent, we must add a term

to the twisted superpotential of the dual theory reflecting an instanton correction in the

original theory [18]. The full dual action is

Sd = − 1

16π

∫d2xd4θ (Y + Y ) log(Y + Y )− 1

8π

∫d2xdθ+dθ−

[Y Σ + µe−Y

]+ c.c.. (B.27)

45

C Central Charge and Anomaly Details

The leading singularities in the OPE of the basic fields of the model (2.22) are:

φi(x)φj(0) ∼ −δij log(x2), ψi,±(x)ψj,±(0) ∼ − iδij

x±±,

pα(x)pβ(0) ∼ −δαβbα

log(x2), ηα,±(x)ηβ,±(0) ∼ − iδαβ

bαx±±, (C.1)

yµ(x)yν(0) ∼ −bµδµν log(x2), χµ,±(x)χν,±(0) ∼ −ibµδµν

x±±,

σa(x)σb(0) ∼ − e2a

2πδab log

(x2), λa,±(x)λb,±(0) ∼ − e

2a

2π

iδabx±±

.

The bottom component of the classical supercurrent J 0−− includes the composite operator∑

i ψi,−ψi,−(x), which we define via point-splitting∑i

ψi,−ψi,−(x) := limy→x

(∑i

ψi,−(y)ei∫ xy QiaAaψi,−(x)− i

(x− y)−−

)(C.2)

=:∑i

ψi,−ψi,−(x) : −∑i

QiaAa−−(x)−∑i

QiaAa++(x) limy→x

(x− y)++

(x− y)−−.

The anomaly in the supercurrent is determined by this operator:

D−J 0−−∣∣ = − 1

4π

[Q+,

∑i

ψi,−ψi,−(x)

]= −√

2

4π

∑i

Qiaλa,−(x) =

∑iQia

4πD−Σa

∣∣ . (C.3)

Thus, the anomaly

γa =∑i

Qia. (C.4)

We determine the central charge of the N = 2 Virasoro algebra generated by J−− +

F−− by considering the leading singularity of the current-current OPE. The R-current, the

bottom component of the superfield is

j−− =iαi4πφiD−−φi +

iγα4π

(D−−pα −D−−pα)− iγµ

4π√bµ∂−− (yµ − yµ)− i

2e2a

σa∂−−σa (C.5)

− 1− αi4π

ψi,−ψi,− −bα4πηα,−ηα,− +

1

4πbµχµ,−χµ,−.

Using the OPEs from above, we see that

j−−(x)j−−(0) ∼ −

(∑i (1− 2αi)−NU(1) +NP +NY + 2

∑αγ2αbα

+ 2∑

µ

γ2µbµ

)(x−−)2

+ . . . , (C.6)

= − c/3

(x−−)2+ . . . .

Hence the quoted formula in the main body of the text (2.32).

46

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